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In this post I will discuss my own understanding of the spectral sparsification paper by Daniel Spielman and Nikhil Srivastava (STOC 2008). I will assume the following:

- The reader is a beginner, like me, and have already glanced through the Spielman-Srivastava paper (from now on, the SS paper).
- The reader has, like me, a basic understanding of spectral sparsification and associated concepts of matrix analysis. I will assume that she has read and understood the Section 2 (Preliminaries) of the SS paper.
- The reader holds a copy of the SS paper while reading my post.

First, I will mention the main theorems (actually, I will mention only what they “roughly” say).